Finite Volume Solution of 2D and 3D Euler and Navier-Stokes ...

176 J. Fürst, M. Janda, and K. KozelLet us consider a general one-dimensional method of the formu n+1i = u n i − C i− 12 (un i − u n i−1) + D i+ 12 (un i+1 − u n i ), (9)then the following theorem of Harten [15] can be used to check the TVD property.Lemma 2.2 (Harten,1983). Let the following conditions be fulfilled ∀i ∈ Z:C i− 1 ≥ 0, D 2 i+ 1 ≥ 0, C 2 i+ 1 + D 2 i+ 1 ≤ 1. (10)2Then the numerical method (9) is TVD.This lemma gives some hints how to correct some classical high order methodsin order to be TVD. Unfortunately, this lemma is valid only for the one-dimensionalcase. In fact, Goodman and LeVeque show in [12], that the TVD property in themultidimensional case implies only a first order accuracy.In spite of their result, many methods based on one-dimensional high orderTVD methods have been constructed for practical multidimensional problems.Although they are not TVD, they remain high order for smooth solutions andusually do not generate oscillations near discontinuities.For practical computations we use the MacCormack predictor-corrector schemebecause of its simple implementation especially for nonlinear systems. The TVDMacCormack scheme has for one-dimensional scalar problem the following form:with G ± defined byandu n+ 1 2i = u n i − ∆t (f(un∆x i ) − f(u n i−1) ) , (11)u n+1i = 1 [u n i + u n+ 1 2i − ∆t (f(u n+ 1 2i+12∆x) − 1 f(un+ 2i )) ] , (12)u n+1i = u n+1i + [ G + (r + i ) + G− (r − i+1 )] (u n i+1 − u n i ) −− [ G + (r + i−1 ) + G− (r − i )] (u n i − u n i−1) (13)G ± (r ± i ) = |f ′ (u n i )|∆t2∆x2.2. Extension to the 2D scalar case(1 − |f ′ (u n i )|∆t ) [ 1 − Φ(r ± i∆x)] (14)Φ(r ± i ) = max ( 0, min(2r ± i , 1)) . (15)The scalar initial value problem for the 2D problem is given by the equationu t + f(u) x + g(u) y = 0 with the initial condition u(x, y, 0) = u 0 (x, y). We againsolve this problem in the unbounded domain (x, y, t) ∈ R × R × R + .For a two-dimensional scalar case we analyzed in [9] a class of monotoneschemes (especially the so called l 1 -contractive schemes) and published in [11] thefollowing lemma:

178 J. Fürst, M. Janda, and K. Kozeland for each k ≠ 0 (we set ˜f k = 0 and ˜f k = 0 for k < −p and k > q)˜f k (u n i−p,j, ..., v n i+q,j) ≤ ˜f k+1 (u n i−p−1,j, ..., v n i+q−1,j), (23)˜g k (u n i,j−p, ..., v n i,j+q) ≤ ˜g k+1 (u n i,j−p−1, ..., v n i,j+q−1) (24)then the schemeu n+1i,j = u n i,j − ∆t (˜fn∆xi+ 1 2 ,j − ˜f)ni− 1 2 ,j − ∆t ()˜g n∆yi,j+− ˜g n 12 i,j−, (25)12˜f n i+ 1 2 ,j = ˜f(u n i−p,j, u n i−p+1,j, ..., u n i+q,j), ˜g n i,j+= ˜g(u n i,j−p, u n i,j−p+1, ..., u n i,j+q)12is monotone (and hence TVD).Unfortunately it is not possible to construct a high order TVD scheme formultidimensional problem [12]. The uniform TVD bound is too restrictive andtherefore one should consider following approach analyzed for example by Coqueland LeFloch in [4, 3]. Let us assume a two-dimensional conservative scheme of theform:u n+1i,j= u n i,j − ∆t (˜fn∆xi+ 1 2 ,j − ˜f)ni− 1 2 ,j − ∆t ()˜g n∆yi,j+− ˜g n 12 i,j−, (26)12with ∆x = ∆y = const.∆t and consistent with the equation u t +f(u) x +g(u) y = 0.Theorem 2.5 (Coquel, LeFloch 1991). Let the numerical fluxes f n i+ 1 2 ,j and gn i,j+ 1 2can be split onto two parts f n i+ 1 2 ,j = pn ∆x+i+ 1 2 ,j ∆t an i+ 1 2 ,j and gn = q n +i,j+ 1 2 i,j+ 1 2with p and q being monotone fluxes (e.g. first order Lax-Friedrichs type)∆y∆t bn i,j+ 1 2and a and b being antidiffusive high order corrections. If there exist for a finitetime T > 0 constants C > 0 and M > 0 independent of ∆t (and consequentlyindependent of ∆x and ∆y) such that||u n i,j|| ∞ < C, |a n i+ 1 ,j| < M∆xα , |b n2 i,j+| < M∆y α , (27)12with n∆t < T and α ∈ ( 2 3, 1), then the piecewise constant function with the valuesgiven by u n i,j converges to the unique entropy solution when ∆t → 0.Example 2.6 (2D version of the scheme proposed by Davis). Let us suppose the 2Dscheme for the two-dimensional linear equation proposed by Davis u t +cu x +du y =0 :u n+1i,j = u n i,j − c ∆t2∆x (un i+1,j − u n i−1,j) + c 2 ∆t22∆x 2 (un i+1,j − 2u n i,j + u n i−1,j) ++ K(r + i,j , r− i+1,j )(un i+1,j − u n i,j) − K(r + i−1,j , r− i,j )(un i,j − u n i−1,j) − (28)−d ∆t2∆y (un i,j+1 − u n i,j−1) + d 2 ∆t22∆y 2 (un i,j+1 − 2u n i,j + u n i,j−1) ++ K(s + i,j , s− i,j+1 )(un i,j+1 − u n i,j) − K(s + i,j−1 , s− i,j )(un i,j − u n i,j−1) (29)

180 J. Fürst, M. Janda, and K. Kozelwhich is a set of m independent scalar problems. For each component we can usethe TVD MacCormack scheme defined in the previous section. In order to get theTVD MacCormack scheme for the original variables W , we multiply the schemewritten for the characteristic variables V by the matrix R. Finally, we getW n+ 1 2i = Wi n − ∆t (AWn∆x i − AWi−1n ), (44)W n+1i = 1 [Wi n + W n+ 1 2i − ∆t () ]AW n+ 1 2i+1 − AW n+ 1 2i , (45)2∆x[W n+1i = W n+1i + R ˜G+ (˜r + i ) + ˜G]− (˜r − i+1 ) R −1 (Wi+1 n − Wi n ) −[−R ˜G+ (˜r + i−1 ) + ˜G]− (˜r − i ) R −1 (Wi n − Wi−1). n (46)Here ˜r ± iare vectors with m components(˜r + i )(l) = ( R −1 (W n i − W n i−1) ) (l)/(R −1 (W n i+1 − W n i ) ) (l), (47)(˜r − i )(l) = ( R −1 (W n i+1 − W n i ) ) (l)/(R −1 (W n i − W n i−1) ) (l), (48)where r (l) denotes the l-th component of the vector r.The viscosity coefficients ˜G are m × m diagonal matrices with the elementsgiven by˜G ± (˜r ± i )(l,l) = 1 |a (l) |∆t2 ∆x(1 − |a(l) |∆t∆x )[1 − Φ((˜r± i )(l) )]. (49)In order to avoid the evaluation of eigenvectors of the Jacobian matrix A weuse the so called simplified TVD scheme proposed by D. M. Causon in [2] whichis for the case of the one-dimensional nonlinear system written as:W n+ 1 2i = Wi n − ∆t (F (Wn∆x i ) − F (Wi−1) n ) , (50)W n+1i = 1 [Wi n + W n+ 1 2i − ∆t (F (W n+ 1 2i+12∆x) − F (W n+ 1 2i )) ] , (51)]W n+1i = W n+1i +[G + (r + i ) + G− (r − i+1 ) (Wi+1 n − Wi n ) −]−[G + (r + i−1 ) + G− (r − i ) (Wi n − Wi−1) n (52)with G ± and r ± given by the formulasr + i = , r− i = ,G ± (r ± i ) = 1 2 C(ν i) [ {1 − Φ(r ± i )] νi (1 − ν, C(ν i ) =i ), ν i ≤ 0.50.25, ν i > 0.5,∆tν i = ρ Ai ∆x . (53)Here < ., . > denotes the standard inner (scalar) product in R m and ρ Ai is thespectral radius of the Jacobi matrix ∂F/∂W at the point W i (for the case of the

182 J. Fürst, M. Janda, and K. KozelFor the two-step LW scheme one has:W n+1/2i = 1 33∑Wk n − ∆t 3∑(F2µk,i∆y n k − G n k,i∆x k ), (63)ik=1W n+1i = Wi n − ∆t 3∑µ ik=1(F n+1/2k,ik=1∆y k − G n+1/2k,i∆x k ). (64)The cell-vertex corresponding to the one-dimensional staggered version hasthe following form (assuming for simplicity that six triangles meet at each vertex):LF:W ∗ i = 1 3W n+10,k= 1 63∑W0,k n − ∆t 3∑(2µ ˜F k ∆y k − ˜G k ∆x k ), (65)ik=1k=1˜F k = 1 2 (F 0,k+1 − F 0,k ), ˜Gk = 1 2 (G 0,k+1 − G 0,k ),6∑Wi ∗ − ∆t 6∑(2µ ˜F i ∗ ∆y i − ˜G ∗ i ∆x y ), (66)0,ki=1i=1˜F ∗ i = 1 2 (F ∗ i+1 − F ∗ i ), ˜G∗ i = 1 2 (G∗ i+1 − G ∗ i ).LW scheme uses the same first step as the above described LF scheme and thesecond step is of the form:W n+10,k= W0,k n − ∆t 6∑(2µ ˜F i ∗ ∆y i − ˜G ∗ i ∆x y ). (67)0,kHere W 0,k are located at the vertices V 0,k of the triangle T i and Wi∗ are locatedat the centers of gravity P i of triangles T i .The composite scheme consists again as N − 1 steps of LW scheme followedby one LF step. The value of N is determined by numerical testing. The cell vertexscheme was also extended to three-dimensional case.Remark: Composite schemes were introduced by R. Liska and B. Wendroff fora 2D nonlinear hyperbolic problem solved by finite difference methods on an orthogonalgrid. The idea is following: in many high order schemes one has to addsome artificial viscosity terms in order to achieve solution without oscillations nearshock waves or steep gradients. Composite schemes use certain number of a highorder scheme (with low value of artificial viscosity) and the arising oscillations arethen smeared by one step of a more dissipative scheme (for example Lax-Friedrichsscheme).We extended the original finite difference schemes to 2D and 3D finite volumemethod of the cell centered or cell vertex form [14].i=1

Solution of 2D and 3D Euler and Navier-Stokes Equations 1834. Implicit finite volume method for 2D inviscid flowsThe numerical solution is again obtained by the finite volume approach: The domainΩ is approximated by a polygonal domain Ω h and this polygonal domain isdivided into m polygonal convex cells 3 C i possess the following property:m⋃Ω h = C i and C i ∩ C j = ∅ for i ≠ j .i=1Figure 1 shows a sample of such adomain divided into 13 triangular, quadrilateral,and pentagonal cells. Let m i denotethe number of cells adjacent to C i (i.e.number of cells that share an edge with thecell C i ) and let the set N i = {i 1 , i 2 , ..., i mi }contain their indices (see Fig. 1 where m i =5). Next, let us denote by Biinlet the set ofedges shared by the cell C i and the inletboundary Γ inlethof Ω h ; similarly for Bioutleti 3Si 3SSii 4 i 24ii 5 S i 1Si1i 5Figure 1. Unstructured gridwith mixed type of cells.and B walli .The basic finite volume scheme is obtainedin the usual way: integrating the conservation law in a cell C i , applyingGreen’s theorem and approximating the integral over the boundary of C i by thenumerical flux functions. The scheme is thenW n+1i = W n i − ∆tR 1 (W n ) i . (68)Here Win stands for the approximation of the solution in the cell C i at a timet = n∆t and R 1 (W n ) i is the component of the residual vector computed as⎡⎤R 1 (W n ) i = 1 ⎣ ∑H(Wi n , Wj n , Sµ(C i )⃗ i,j ) + ∑ ∑H b (Wi n , S ⃗ e ) ⎦ . (69)j∈N i bThe superscript 1 denotes the first order approximation, µ(C i ) is the volume ofthe cell C i , ⃗ S i,j denotes the outer normal vector to the common edge between C iand C j , the function H is the numerical flux, b denotes the type of the boundaryconditions and belongs to the set b ∈ {inlet, outlet, wall}, H b is the numerical fluxthrough the boundary and ⃗ S e denotes the outer normal vector to the boundaryedge e. Both vectors ⃗ S i,j and ⃗ S e have the length equal to the length of the correspondingedge. The numerical flux H(W n i , W n j , ⃗ S i,j ) in the previous formula isthe numerical approximation of the integral of the physical flux function over thee∈B b ii 23 Since the structured grid can be viewed from the mathematical point of view as a special caseof the unstructured grid, we present only the scheme for unstructured meshes.

184 J. Fürst, M. Janda, and K. Kozelcommon edge e i,j shared between C i and C j :⎛⎛⎞ ⎛∫ρuH(Wi n , Wj n , S ⃗ i,j ) ≈ ⎜⎜ρu 2 + p⎟⎝⎝e i,jρuv ⎠ n x + ⎜⎝(e + p)uρvρuvρv 2 + p(e + p)v⎞⎟⎠ n y⎞⎟⎠ dS (70)where n x and n y are the components of the unit normal vector to the edge e i,joriented as the outer normal for the cell C i . Analogous, H b (W n i , ⃗ S e ) is the approximationof the flux through the edge on the boundary.4.1. First order implicit schemeAs a building block for the implicit scheme we choose the first order finite volumescheme based on Osher’s flux and the related approximate Riemann solver (see[16]). The advantage of the Osher flux is that one can evaluate simply the Jacobiansof the numerical flux function which are needed for the implicit scheme.The usual explicit first order scheme is thenW n+1 = W n − ∆tR 1 (W n ) (71)and the implicit scheme is obtained from the explicit version (68) by replacingR 1 (W n ) by R 1 (W n+1 ):W n+1i = W n i − ∆tR 1 (W n+1 ) i . (72)The operator R is nonlinear, so we can’t solve this equation directly. Therefore welinearize the equation at the point W n :W n+1 = W n − ∆t(R 1 (W n ) + ∂R1 (W n+1 − W n)) , (73)∂Whence,( ) I∆t + ∂R1 (W n+1 − W n) = −R 1 (W n ). (74)∂WThe matrix ∂R1∂Wis evaluated at the point W n using the expressions for the Jacobianmatrices of Osher’s flux functions ∂H(WL,WR,⃗ S)∂W Land ∂H(WL,WR,⃗ S)∂W Rand theappropriate Jacobian matrices of boundary fluxes.The resulting system of linearized equations is solved using a GMRES methodpreconditioned with the ILU decomposition.4.2. Second order semi-implicit schemeIn order to improve the accuracy of the basic first order scheme we use a piecewiselinear reconstruction of the solution (for details see [10]). The second order semiimplicitscheme uses the high order residual R 2 on the right hand side (explicitpart) and the matrix on the left hand side is computed from the low order residual:( ) I∆t + ∂R1 (W n+1 − W n) = −R 2 (W n ). (75)∂W

Solution of 2D and 3D Euler and Navier-Stokes Equations 185The second order residual R 2 is computed in the following way: we computefirst the approximation of the gradient of the solution gradW n in each cell 4 andthen using this gradient we define the second order residual vector byW L i,j = W n i + (⃗x i,j − ⃗x i ) · gradW n i , (76)Wi,j R = Wj n + (⃗x j,i − ⃗x j ) · gradWj n ,⎡R 2 (W n ) i =1⎣ ∑H(Wµ(C i )i,j, L Wi,j, R S ⃗ i,j ) + ∑j∈N i b⎤∑H b (Wi,j, L S ⃗ e ) ⎦ ,where ⃗x i is the center of gravity of the cell i and ⃗x i,j is the center of the commonedge between the cells i and j. The numerical fluxes H and H b are computed inthe same way as for the first order scheme (i.e. using Osher’s Riemann solver) butinstead of W i and W j we use the interpolated values W R i,j and W L i,j .In the steady case one has R 2 (W n ) = 0 and therefore the scheme is high order.Moreover, as the second order residual is of ENO type, the stationary solution isessentially non-oscillatory. In the unsteady case the scheme is only low order dueto replacing R 2 by R 1 in the implicit part.e∈B b i5. 2D transonic inviscid flow through a channel and a turbinecascade5.1. Transonic flow through the 2D test channel with a bumpAs a first test case we choose the transonic flow through the two-dimensional testchannel with a bump, i.e. the so-called Ron-Ho-Ni channel. This is a well-knowntest case and it was solved by many researchers. See for example [5], [6].1.4Causon’s schemeFull TVD scheme1.4Implicit scheme1.21.211Mach number0.8Mach number0.80.60.60.40.4-1 -0.5 0 0.5 1 1.5 2x-1 -0.5 0 0.5 1 1.5 2x(a) TVD MacCormackscheme(b) Implicit scheme(c) Composite schemeFigure 2. Distribution of the Mach number along the lower andupper walls for the 2D channel.4 The evaluation of the gradients is described in in detail [10].

186 J. Fürst, M. Janda, and K. KozelWe use the structured mesh with 120 × 30 quadrilateral cells for the TVDMacCormack scheme, an unstructured triangular mesh with 4424 triangles (withrefinement in the vicinity of the shock wave) for the implicit scheme and a structuredtriangular mesh with 10800 triangles for the composite scheme. At the inlet(x = −1) we prescribe the stagnation pressure p 0 = 1, the stagnation densityρ 0 = 1 and the inlet angle α 1 = 0. At the outlet we keep the pressure p 2 = 0.737.The upper (y = 1) and lower part are solid walls.Figure 2(a) shows the distribution of the Mach number along the upperand the lower wall after 30000 iterations of two above mentioned variants of theMacCormack scheme with CF L = 0.5 while figure 2(b) shows the results obtainedby the implicit scheme and figure 2(c) by the composite scheme.We can see that the results obtained by the full TVD MacCormack schemeare quite good. Causon’s simplified scheme uses too much artificial dissipation.5.2. Transonic flow through the 2D turbine cascade SE 1050Next we solve the transonic flow through the 2D turbine cascade SE 1050 givenby Škoda Plzeň. Figure 3 shows the results of the interferometric measurementobtained for the inlet Mach number M 1 = 0.395 [17]. One can see the characteristicstructure of the shock waves emitted from the outlet edge and the reflected shockwaves. Moreover, one can notice the recompression zone on the suction side ofthe blade. Our computation was performed on a structured mesh with 200 × 40quadrilateral cells for the full TVD MacCormack scheme (see Fig. 4(a)) and on anunstructured mesh with 7892 triangles for the implicit scheme (Fig. 4(b)).Figure 3. Interferometric measurement of SE 10505.3. Laminar viscous flow through a 2D turbine cascadeNext we solve the transonic viscous flow through the DCA 8% cascade. We considerthe flow with the non-dimensional viscosity µ = 10 −4 which gives, for the inletconditions p 0 = 1, ρ 0 = 1, α 1 = 2 o and outlet pressure p 2 = 0.48, the value of

Solution of 2D and 3D Euler and Navier-Stokes Equations 187(a) TVD MacCormack scheme,structured grid(b) Implicit scheme, unstructuredgridFigure 4. Distribution of the Mach number in a 2D turbine cascadethe Reynolds number Re = 6450, inlet Mach number M 1 = 0.76 and outlet Machnumber M 2 = 1.03.We use a simple structured mesh with 90 × 50 quadrilateral cells refinedin the vicinity of profiles. Figure 5 shows the results obtained by an improvedversion of Causon’s scheme 5 after 50000 and 50200 iterations. We can see thatthe solution is non-stationary (see the changes in the shape of the wake). Similarnon-stationary solution was obtained also by using a finite difference ENO scheme[1]. Let us mention that these flow conditions (it means relatively low Reynoldsnumber and high Mach number) are not interesting for practical applications. Wehave done this computation in order to show that the effects of artificial viscositycan be very important for a viscous flow calculation. As a matter of fact, a similarcase was formerly solved by M. Huněk and K. Kozel [8] using an implicit residualaveraging method. Their method gave stationary results whereas our computationleads to a non-stationary solution. This is probably due to the fact, that theresidual smoothing method uses too much artificial viscosity.5 The so called modified scheme which we published for example in [11].

188 J. Fürst, M. Janda, and K. Kozel(a) 50000 time steps(b) 50200 time stepsFigure 5. Isolines of Mach number for the non-stationary laminartransonic flow through the DCA 8% cascade.6. 3D inviscid transonic flows6.1. 3D transonic flow through a channelThe 2D composite scheme of cell-vertex type was extended to 3D case using thefollowing finite volume mesh given by fig. 6. Figure 6(a) shows basic finite volume,fig. 6(b) shows the dual finite volume and fig. 6(c) shows the global 3D basic mesh.We used 180×30×12 mesh and 3D channel was changed by bump with the height(a) Basic cell (b) Dual cell (c) Global meshFigure 6. 3D finite volume mesh for composite schemes.equal to 0.18 at z = 0 decreasing to 0.10 at z = 1. Figure 7 shows Mach numberdistribution in the planes z = const.. Figure 7(a) shows the solution at the wallwith the bump (with the shock wave) and smooth solution at the upper wall.Figure 7(b) shows the change of the strength of the shock wave for z between 0and 1. Here we use 25 steps of LW scheme followed by 1 LF step.6.2. 3D transonic flow through a turbine cascadeThe three-dimensional Causon’s scheme and its improved variant are used for thecomputation of the transonic flow through the stator stage of the real 3D turbinegiven by Škoda Plzeň company.

Solution of 2D and 3D Euler and Navier-Stokes Equations 189(a) Mach number along the walls(b) Mach number alongthe lower wall (projections)Figure 7. Mach number distribution in 3D channelAt the inlet we prescribe the stagnation pressure p 0 (r) = 0.38274, stagnationdensity ρ 0 (r) = 1. The direction of the velocity at the inlet is given by two anglesα 1 (r) and µ 1 (r).We use a structured mesh with 90 × 24 × 17 hexahedral cells. Figure 8 showsdistribution of the Mach number obtained by the above mentioned modificationof 3D Causon’s scheme. Figures 9(a)-10(b) show the distribution of the Machnumber for different section and on the pressure and suction side of the blade.Similar results were obtained by J. Fořt and J. Halama [7] by using a cell vertexscheme of Ni.References[1] Philippe Angot, Jiří Fürst, and Karel Kozel. TVD and ENO schemes for multidimensionalsteady and unsteady flows. a comparative analysis. In Fayssal Benkhaldounand Roland Vilsmeier, editors, Finite Volumes for Complex Applications. Problemsand Perspectives, pages 283–290. Hermes, july 1996.[2] D. M. Causon. High resolution finite volume schemes and computational aerodynamics.In Josef Ballmann and Rolf Jeltsch, editors, Nonlinear Hyperbolic Equations- Theory, Computation Methods and Applications, volume 24 of Notes on NumericalFluid Mechanics, pages 63–74, Braunschweig, March 1989. Vieweg.[3] Frédéric Coquel and Philippe Le Floch. Convergence of finite difference schemesfor conservation laws in several space dimensions: the corrected antidiffusive fluxapproach. Mathematics of computation, 57(195):169–210, july 1991.[4] Frédéric Coquel and Philippe Le Floch. Convergence of finite difference schemes forconservation laws in several space dimensions: a general theory. SIAM J. Numer.Anal., 30(3):675–700, June 1993.[5] Vít Dolejší. Sur des méthodes combinant des volumes finis et des éléments finis pourle calcul d’ecoulements compressibles sur des maillages non structurés. PhD thesis,L’Université Méditerranée Marseille et Univerzita Karlova Praha, 1998.

190 J. Fürst, M. Janda, and K. KozelYZXFigure 8. Mach number distribution in the 3D turbine.[6] Miloslav Feistauer, Jiří Felcman, and Mária Lukáčová-Medviďová. Combined finiteelement-finite volume solution of compressible flow. Journal of Computational andApplied Mathemetics, (63):179–199, 1995.[7] J. Fořt, J. Halama, A. Jirásek, M. Kladrubský, and K. Kozel. Numerical solution ofseveral 2d and 3d internal and external flow problems. In R. Rannacher M. Feistauerand K. Kozel, editors, Numerical Modelling in Continuum Mechanics, pages 283–291,September 1997.[8] Jaroslav Fořt, Miloš Huněk, Karel Kozel, J. Lain, Miroslav Šejna, and MiroslavaVavřincová. Numerical simulation of steady and unsteady flows through plane cascades.In S. M. Deshpande, S. S. Desai, and R. Narasimha, editors, Fourteenth InternationalConference on Numerical Methods in Fluid Dynamics, Lecture Notes inPhysics, pages 461–465. Springer, 1994.[9] Jiří Fürst. Modern difference schemes for solving the system of Euler equations.Diploma thesis, Faculty of Nuclear Science and Physical Engineering, CTU Prague,1994. (in czech).[10] Jiří Fürst. Numerical modeling of the transonic flows using TVD and ENO schemes.PhD thesis, ČVUT v Praze and l’Universit de la Mditerrane, Marseille, 2000. inpreparation.[11] Jiří Fürst and Karel Kozel. Using TVD and ENO schemes for numerical solutionof the multidimensional system of Euler and Navier-Stokes equations. In PitmanResearch Notes, number 388 in Mathematics Series, 1997. Conference on Navier-Stokes equations, Varenna 1997.

Solution of 2D and 3D Euler and Navier-Stokes Equations 191(a) Hub (k = 1) (b) Middle (k = 9)(c) Tip (k = 18)Figure 9. Distribution of Mach number for the sections k = const.

192 J. Fürst, M. Janda, and K. Kozel(a) Mach number distribution onthe pressure side of the blade(b) Mach number distribution onthe suction side of the bladeFigure 10. Distribution of Mach number on the blade[12] J.B. Goodman and R.J. LeVeque. On the accuracy of stable schemes for 2D scalarconservation laws. Math. Comp., 45:503–520, 1988.[13] Ami Harten. High resolution schemes for hyperbolic conservation laws. Journal ofComputational Physics, 49:357–393, 1983.[14] Michal Janda, Karel Kozel, and Richard Liska. Composite schemes on triangularmeshes. In Proceedings of HYP 2000, Magdeburg, March 2000. to appear.[15] Randall J. Le Veque. Numerical Methods for Conservation Laws. Birkhäuser Verlag,Basel, 1990.[16] Stanley Osher and Sukumar Chakravarthy. Upwind schemes and boundary conditionswith applications to Euler equations in general geometries. J. Comp. Phys.,(50):447–481, 1983.[17] M. Šťastný and P. Šafařík. Experimental analysis data on the transonic flow past aplane turbine cascade. ASME Paper, (91-GT-313), 1990.Acknowledgment: This work was supported by grants No. 101/98/K001, 201/99/0267 ofGAČR and by the Research Plan MSM 98/210000010.Department of Technical Mathematics,Faculty of Mechanical Engineering,Czech Tecnical University,Karlovo nám. 13121 35, Praha, Czech RepublicE-mail address: furst@marian.fsik.cvut.cz